Primality proof for n = 11549194661:

Take b = 2.

b^(n-1) mod n = 1.

9466553 is prime.
b^((n-1)/9466553)-1 mod n = 2204506690, which is a unit, inverse 9124742838.

(9466553) divides n-1.

(9466553)^2 > n.

n is prime by Pocklington's theorem.